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In examining the dédinition a $2$-category link text, $3$-category link text , $n$-category link text and $(m, n)$-category link text. Can we define an $\alpha $-category, where $\alpha \in \mathbb{K}$? $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$.

Can we define a $M$-category where $M \in GL(\mathbb{K})$? \, $\mathbb{K}=\mathbb{R} $ or $\mathbb{C}$.

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For example, what about $\frac{1}{2}$-category or $\pi$-category? – Labaïr Oct 26 2011 at 7:13
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 What for? What would be the motivation? What motivated you to think about this? – Fernando Muro Oct 26 2011 at 8:03
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 Huh. What have you tried? You might consider defining first an $\alpha$-homotopy group, but for what purpose I don't know. – Todd Trimble Oct 26 2011 at 10:12
The answer is "yes". Here's a definition. An $\alpha$-category is an $\infty$-category such that all $k$-morphisms are identities for $k > \operatorname{Re}(\alpha)$. Unfortunately, it isn't any more useful than existing notions. Are you seeking a specific property that isn't satisfied by this definition? – S. Carnahan♦ Oct 26 2011 at 14:48
There's the idea of homotopy types of spectra as $\mathbb{Z}$-groupoids: math.ucr.edu/home/baez/week121.html. – David Corfield Oct 26 2011 at 15:08
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closed as not a real question by Bruce Westbury, Todd Trimble, Dan Petersen, S. Carnahan♦ Oct 28 2011 at 2:11

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